| L(s) = 1 | − 3·3-s + 3·5-s + 14·7-s − 3·9-s + 15·11-s − 9·15-s + 51·17-s − 27·19-s − 42·21-s − 9·23-s − 19·25-s + 18·27-s − 12·29-s + 21·31-s − 45·33-s + 42·35-s − 31·37-s − 20·43-s − 9·45-s − 75·47-s + 147·49-s − 153·51-s + 57·53-s + 45·55-s + 81·57-s + 141·59-s − 141·61-s + ⋯ |
| L(s) = 1 | − 3-s + 3/5·5-s + 2·7-s − 1/3·9-s + 1.36·11-s − 3/5·15-s + 3·17-s − 1.42·19-s − 2·21-s − 0.391·23-s − 0.759·25-s + 2/3·27-s − 0.413·29-s + 0.677·31-s − 1.36·33-s + 6/5·35-s − 0.837·37-s − 0.465·43-s − 1/5·45-s − 1.59·47-s + 3·49-s − 3·51-s + 1.07·53-s + 9/11·55-s + 1.42·57-s + 2.38·59-s − 2.31·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.702740381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.702740381\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p T + 4 p T^{2} + p^{3} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 28 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 15 T + 104 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 - p T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + 27 T + 604 T^{2} + 27 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T - 448 T^{2} + 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 21 T + 1108 T^{2} - 21 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 31 T - 408 T^{2} + 31 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 75 T + 4084 T^{2} + 75 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 57 T + 440 T^{2} - 57 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 141 T + 10108 T^{2} - 141 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 141 T + 10348 T^{2} + 141 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 49 T - 2088 T^{2} + 49 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 126 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 45 T + 6004 T^{2} + 45 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 73 T - 912 T^{2} + 73 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13586 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 99 T + 11188 T^{2} - 99 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 18050 T^{2} + p^{4} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.85812791908300495274899568142, −13.11636990209035990742086798502, −12.12494265750984570131212651931, −12.10763757215575182759147965683, −11.57517301898252890784134137938, −11.27308501690876625803415233588, −10.41290321489426471643210572794, −10.23207693595868898588502446850, −9.439484184654691738400925917574, −8.743126392890008527275074788892, −8.040030627529234577788485249960, −7.87601704268262659033521622578, −6.82718351490688586277730332877, −6.17065364639748473024377254691, −5.45153242023210553270722603753, −5.35086373085198882901340646873, −4.35102810335025232988051383997, −3.55227920121510277483970729930, −1.99827174172816095797146914257, −1.18224784871517366181068520123,
1.18224784871517366181068520123, 1.99827174172816095797146914257, 3.55227920121510277483970729930, 4.35102810335025232988051383997, 5.35086373085198882901340646873, 5.45153242023210553270722603753, 6.17065364639748473024377254691, 6.82718351490688586277730332877, 7.87601704268262659033521622578, 8.040030627529234577788485249960, 8.743126392890008527275074788892, 9.439484184654691738400925917574, 10.23207693595868898588502446850, 10.41290321489426471643210572794, 11.27308501690876625803415233588, 11.57517301898252890784134137938, 12.10763757215575182759147965683, 12.12494265750984570131212651931, 13.11636990209035990742086798502, 13.85812791908300495274899568142
